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DTSTART:20070311T020000
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DTSTAMP:20260125T172305
DTSTART;TZID=America/Detroit:20260127T150000
DTEND;TZID=America/Detroit:20260127T160000
SUMMARY:Workshop / Seminar:Derived Fun: Towards Homotopical Algebra
DESCRIPTION:Derived functors play a central role in commutative algebra\, organizing algebraic computations through exact sequences and homological methods. This talk begins with a brief conceptual revisit of derived functors in homological algebra\, and then examines Kähler differentials of commutative rings as a motivating example for why we want derived functors beyond abelian categories. \n\nTo address this problem\, we trace Dan Quillen’s insight that homotopical methods provide an ideal framework for defining derived functors in nonabelian settings. Along the way\, we encounter his definition of model categories\, which have since become central tools in modern homotopy theory. From this perspective\, the cotangent complex arises as a derived replacement for Kähler differentials. We will outline the construction of the cotangent complex and discuss some of its powerful applications to deformation theory. Time permitting\, I will say something about how the homotopical viewpoint reconciles with homological algebra.
UID:144446-21895363@events.umich.edu
URL:https://events.umich.edu/event/144446
CLASS:PUBLIC
STATUS:CONFIRMED
CATEGORIES:Mathematics
LOCATION:East Hall - 3088
CONTACT:
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BEGIN:VEVENT
DTSTAMP:20251226T220321
DTSTART;TZID=America/Detroit:20260127T160000
DTEND;TZID=America/Detroit:20260127T170000
SUMMARY:Lecture / Discussion:Colloquium: Topology\, graphs\, and data
DESCRIPTION:This talk will be an introduction to the emerging area of discrete\nhomotopy theory\, which applies intuitions and techniques from the\ncontinuous setting to discrete objects such as graphs. It has found a\nbroad range of applications\, both within and outside mathematics\,\nincluding to matroid theory\, hyperplane arrangements\, and data analysis.\n\nI will discuss two of my own contributions to discrete homotopy theory\,\none more theoretical and one more applied. The first is a proof\, joint\nwith D. Carranza (Compos. Math.\, 2024)\, of the conjecture by E. Babson\,\nH. Barcelo\, M. de Longueville\, and R. Laubenbacher that discrete\nhomotopy groups can be topologically realized. The second\, joint with N.\nKershaw (arXiv:2506.15020)\, builds on this result and introduces a new\nmethod of data analysis\, which we call persistent discrete homology. We\nshow that in addition to its utility for clustering\, it can detect other\ngeometric features of a data set. It is furthermore highly noise\nresistant\, and as such provides a powerful alternative to the usual\nmethods of (unsupervised) machine learning\, especially in areas subject\nto high uncertainty\, such as seismology or crime linkage.
UID:143123-21892177@events.umich.edu
URL:https://events.umich.edu/event/143123
CLASS:PUBLIC
STATUS:CONFIRMED
CATEGORIES:Mathematics
LOCATION:East Hall - 1360
CONTACT:
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